Step |
Hyp |
Ref |
Expression |
1 |
|
setsid.e |
|- E = Slot ( E ` ndx ) |
2 |
|
setsval |
|- ( ( W e. A /\ C e. V ) -> ( W sSet <. ( E ` ndx ) , C >. ) = ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ) |
3 |
2
|
fveq2d |
|- ( ( W e. A /\ C e. V ) -> ( E ` ( W sSet <. ( E ` ndx ) , C >. ) ) = ( E ` ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ) ) |
4 |
|
resexg |
|- ( W e. A -> ( W |` ( _V \ { ( E ` ndx ) } ) ) e. _V ) |
5 |
4
|
adantr |
|- ( ( W e. A /\ C e. V ) -> ( W |` ( _V \ { ( E ` ndx ) } ) ) e. _V ) |
6 |
|
snex |
|- { <. ( E ` ndx ) , C >. } e. _V |
7 |
|
unexg |
|- ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) e. _V /\ { <. ( E ` ndx ) , C >. } e. _V ) -> ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) e. _V ) |
8 |
5 6 7
|
sylancl |
|- ( ( W e. A /\ C e. V ) -> ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) e. _V ) |
9 |
1 8
|
strfvnd |
|- ( ( W e. A /\ C e. V ) -> ( E ` ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ) = ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) ) |
10 |
|
fvex |
|- ( E ` ndx ) e. _V |
11 |
10
|
snid |
|- ( E ` ndx ) e. { ( E ` ndx ) } |
12 |
|
fvres |
|- ( ( E ` ndx ) e. { ( E ` ndx ) } -> ( ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) ` ( E ` ndx ) ) = ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) ) |
13 |
11 12
|
ax-mp |
|- ( ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) ` ( E ` ndx ) ) = ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) |
14 |
|
resres |
|- ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) = ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) |
15 |
|
incom |
|- ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) = ( { ( E ` ndx ) } i^i ( _V \ { ( E ` ndx ) } ) ) |
16 |
|
disjdif |
|- ( { ( E ` ndx ) } i^i ( _V \ { ( E ` ndx ) } ) ) = (/) |
17 |
15 16
|
eqtri |
|- ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) = (/) |
18 |
17
|
reseq2i |
|- ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) = ( W |` (/) ) |
19 |
|
res0 |
|- ( W |` (/) ) = (/) |
20 |
18 19
|
eqtri |
|- ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) = (/) |
21 |
14 20
|
eqtri |
|- ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) = (/) |
22 |
21
|
a1i |
|- ( ( W e. A /\ C e. V ) -> ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) = (/) ) |
23 |
|
elex |
|- ( C e. V -> C e. _V ) |
24 |
23
|
adantl |
|- ( ( W e. A /\ C e. V ) -> C e. _V ) |
25 |
|
opelxpi |
|- ( ( ( E ` ndx ) e. _V /\ C e. _V ) -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) ) |
26 |
10 24 25
|
sylancr |
|- ( ( W e. A /\ C e. V ) -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) ) |
27 |
|
opex |
|- <. ( E ` ndx ) , C >. e. _V |
28 |
27
|
relsn |
|- ( Rel { <. ( E ` ndx ) , C >. } <-> <. ( E ` ndx ) , C >. e. ( _V X. _V ) ) |
29 |
26 28
|
sylibr |
|- ( ( W e. A /\ C e. V ) -> Rel { <. ( E ` ndx ) , C >. } ) |
30 |
|
dmsnopss |
|- dom { <. ( E ` ndx ) , C >. } C_ { ( E ` ndx ) } |
31 |
|
relssres |
|- ( ( Rel { <. ( E ` ndx ) , C >. } /\ dom { <. ( E ` ndx ) , C >. } C_ { ( E ` ndx ) } ) -> ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } ) |
32 |
29 30 31
|
sylancl |
|- ( ( W e. A /\ C e. V ) -> ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } ) |
33 |
22 32
|
uneq12d |
|- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) u. ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) ) = ( (/) u. { <. ( E ` ndx ) , C >. } ) ) |
34 |
|
resundir |
|- ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) = ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) u. ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) ) |
35 |
|
un0 |
|- ( { <. ( E ` ndx ) , C >. } u. (/) ) = { <. ( E ` ndx ) , C >. } |
36 |
|
uncom |
|- ( { <. ( E ` ndx ) , C >. } u. (/) ) = ( (/) u. { <. ( E ` ndx ) , C >. } ) |
37 |
35 36
|
eqtr3i |
|- { <. ( E ` ndx ) , C >. } = ( (/) u. { <. ( E ` ndx ) , C >. } ) |
38 |
33 34 37
|
3eqtr4g |
|- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } ) |
39 |
38
|
fveq1d |
|- ( ( W e. A /\ C e. V ) -> ( ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) ` ( E ` ndx ) ) = ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) ) |
40 |
13 39
|
eqtr3id |
|- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) = ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) ) |
41 |
10
|
a1i |
|- ( ( W e. A /\ C e. V ) -> ( E ` ndx ) e. _V ) |
42 |
|
fvsng |
|- ( ( ( E ` ndx ) e. _V /\ C e. V ) -> ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) = C ) |
43 |
41 42
|
sylancom |
|- ( ( W e. A /\ C e. V ) -> ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) = C ) |
44 |
40 43
|
eqtrd |
|- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) = C ) |
45 |
3 9 44
|
3eqtrrd |
|- ( ( W e. A /\ C e. V ) -> C = ( E ` ( W sSet <. ( E ` ndx ) , C >. ) ) ) |